A Mathematical Breakdown of How Automated Market Makers Price Assets

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Learn how AMMs price tokens using x·y=k, how fees accrue, and why impermanent loss hurts LPs. From Uniswap V2 to V3, the math explained clearly.
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AMMs use the formula x·y=k to price tokens without an order book. Swaps move reserves along a hyperbola, causing price impact on large trades. A 0.30% fee grows k over time, rewarding liquidity providers. LPs face impermanent loss — always worse than holding when price moves, calculated as IL(r) = 2√r/(1+r) − 1. Uniswap V3 improves capital efficiency via concentrated liquidity ranges, while Curve and Balancer extend the model to stablecoins and multi-asset pools.